##\det## is just a homomorphism from the group of linear maps on ##V## to its representation on the top exterior power ##\Lambda^n V##. Taking some ##v_i \in V##, we have ##\det A : \Lambda^n V \to \Lambda^n V## given by

[tex](\det A)(v_1 \wedge \ldots \wedge v_n) = A(v_1) \wedge \ldots \wedge A(v_n)[/tex]
From here it is easy to show ##\det AB = \det A \det B## by using the usual composition of linear maps on each factor in the wedge product on the right.